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A Role for Categories in the Foundations of Quantum Theory

Published online by Cambridge University Press:  28 February 2022

David Geoffrey Holdsworth
David Geoffrey Holdsworth*
Affiliation:
Institut für Theoretische Physik, Universität zu Köln Department of Philosophy, The University of Western Ontario
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Extract

In this paper I shall outline a possibility for incorporating category theory in an essential way into the foundations of quantum theory. The role for category theory which I shall sketch here is strictly foundational in the sense that it offers no new perspectives for a physicist working within the quantum formalism, but rather provides a way of interpreting the algebraic structure by giving an alternative axiomatization of the underlying logic of the theory. The axiomatization is given without recourse to any classical set theory by invoking methods closely connected with the work of F. W. Lawvere. More precisely, then, a possible approach to the foundations of physics which axiomatizes its algebraic structure in a way which is independent of set theory and hence points, as it were, to a structuralist foundation of physical theories, will be proposed.

Type
Part VII. Quantum Logic
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1978 , Issue 1: Volume One: Contributed Papers 1978 , 1978 , pp. 257 - 267
Copyright
Copyright © 1978 by the Philosophy of Science Association

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Footnotes

1

Work done while supported by the Heinrich Hertz Stiftung, W. Germany.

References

Eilenberg, S. and MacLane, S. “General Theory of Natural Equivalences.” Transactions of the American Mathematical Society. 58(1945): 231–294.CrossRefGoogle Scholar
Herrlich, H. and Strecker, G.E. Category Theory. Boston: Allyn and Bacon Inc., 1973.Google Scholar
Holdsworth, D.G. “Category Theory and Quantum Mechanics.” Journal of Philosophical Logic. 6(1977): 441–453.CrossRefGoogle Scholar
Holdsworth, D.G. “An Elementary Theory of the Category of Complete Atomic Boolean Algebras.” Unpublished manuscript.Google Scholar
Lawvere, F.W. “An Elementary Theory of the Category of Sets.” Proceedings of the National Academy of Sciences. 52(1964): 1506–1511. (A more extensive treatment is to be found in a paper by the same title, mimeographed notes, University of Chicago, 1965.)CrossRefGoogle Scholar
Lawvere, F.W. “The Category of Categories as a Foundation for Mathematics.” Proceedings of the Conference on Categorical Algebra, La Jolla. Heidelberg/New York: Springer-Verlag, 1966.Google Scholar
Middlestaedt, P. Quantum Logic. Dordrecht: D. Reidel, forthcoming.Google Scholar
van Fraassen, B.C. “Semantic Analysis of Quantum Logic.” Contemporary Research in the Foundations and Philosophy of Quantum Theory. (The University of Western Ontario Series in Philosophy of Science, Volume 2). Edited by Hooker, C.A. Dordrecht: D. Reidel, 1973. Pages 80–113.Google Scholar